Return is not one predicate. Across five exact finite Pascal-carrier studies, endpoint recurrence, structural recovery, representation recovery, operator inversion, route history and full identity separate. The positive connector is an injective quaternion representation with an exact cyclic intertwiner and inverse conjugation for every finite carrier PN. Nearby negative results show that terminal equality does not recover history, collision-free description need not compress, and algebraic return neither selects a global seam nor guarantees metric isometry.
A Return claim must name its level.
The shared method is not a universal mechanism. It is a typing discipline: declare the carrier, representation, operator, comparison and residual before saying that something “returned”.
Figure 1. Code-native synthesis diagram. Broken implication marks separate logical claim levels.
The quaternion realizes the cyclic three-coordinate action.
For PN={(a,b,c)∈ℤ³≥0:a+b+c≤N}, set Φ(a,b,c)=(a+b+c)1+ai+bj+ck and u=(1+i+j+k)/2. The scalar depth is fixed, while conjugation cycles the imaginary channels.
Adᵤ(i)=j, Adᵤ(j)=k, Adᵤ(k)=i Adᵤ(Φ(a,b,c)) = Φ(c,a,b) Adᵤ⁻¹(Adᵤ(q)) = q
Figure 2. Exact connector. The theorem holds for N≥0; faithful restricted action requires N≥1 because P₀ is a singleton.
Four cases, four different limits.
Endpoint is not history.
100 frozen histories collapse to 56 terminal classes.
Unique address is not compression.
Collision-free signatures required one signature per history.
Return is not seam selection.
(Q′ε)²=εI returns algebraically, but does not select the seam or imply isometry.
Exact inverse is possible when typed.
Injectivity, cyclic equivariance and inverse Return hold with proper/improper and N=0 boundaries explicit.
Use the instruments—without making them evidence.
These governed HTMLs provide visual grammar for record, cut, seam, residual and Return. They do not control the mathematical claims.
Record / Play Operator
Shared readout and operator non-identity.
Open governed HTML ↗Return · Residual · Lock
Point, direction and curvature residuals under tolerance.
Open governed HTML ↗Knot + Rope around Body
Closed route with a directional residual.
Open governed HTML ↗From Gap to Return
Cover, kernel, cut and residual vocabulary.
Open governed HTML ↗Quaternion Return Switchyard
Technical reader behind the exact connector.
Open technical reader ↗Pascal Pentagon Explorer
Finite 106-node quotient and typed D₅ explorer.
Open technical reader ↗The Labreports remain authoritative.
| Study | Result role | State | Source |
|---|---|---|---|
| A0-BL-021 | history / state non-identity | closed | Labreport |
| A0-BL-035 | identification / compression | closed | Labreport |
| A0-BL-045 | operator / seam / metric | closed | Labreport |
| A0-BL-052 | finite connector | accepted | Labreport |
| A0-BL-053 | all-N proof | R2 internal | Labreport |